Massive scalar field quasinormal modes of higher dimensional black holes

We study the quasinormal spectrum of a massive scalar field in a $D$-dimensional black hole background. We found a qualitatively different dependence of the fundamental modes on the field mass for $D\ge 6$. The behavior of higher modes is qualitatively the same for all $D$. Thus for some particular values of $\mu M^{1/(D-3)}$, where $\mu$ is the field mass and $M$ is the black hole mass, the spectrum has two dominating oscillations with a very long lifetime. We show that the asymptotically high overtones do not depend on the field mass. In addition, we present the generalization of the Nollert improvement of the continued fraction technique for the numerical calculation of quasinormal frequencies of $D$-dimensional black holes.

Figure 1

Fundamental quasinormal frequencies ($l=0$) for different $D$ and $\mu$. The frequency for $\mu =0$ has the largest imaginary part. The points were plotted with the step of $\Delta \mu =0.1/r_0$. Solid lines mark the same number of $D$.

Figure 2

Fundamental quasinormal frequencies for $D=6$ as function of $\mu r_0$ for $l=0$, 1, 2 (represented as dots, rhombuses and triangles, respectively). The dashed line corresponds to $\mathrm{Re}(\omega )=\mu$.

Figure 3

Three higher quasinormal modes ($l=0$) for $D=6$ and different $\mu$. The frequency for $\mu =0$ has the largest imaginary part. The points were plotted with the step of $\Delta \mu =1/(2r_0)$. Solid lines mark the same overtone number.

Figure 4

First 20 quasinormal modes for $D=7$, $l=0$, $\mu r_0=0$, 0.2, 0.4, 0.6, 0.8, 1.0.